Computer simulations were done in order to verify the response of the created model. The vehicle Parameters are shown in Table 1. These values are used from, [37]. The code used for the simulation can be seen in 0.
Table 4.1: The numeric constants used for the simulations
MB 50 (kg) IB 5.5 (kg/m2) Mi 3.5 (kg) Ii .025 (kg/m2) Is .009 (kg/m2) W .75 (m) H 1 (m) rw .085 (m)
The model is extremely nonlinear, as shown in the derivation from Chapter 3. However, simple simulations can still be performed, in lieu of a controller, in order to see if the system’s response is realistic. Figure 4.1 shows the velocity response of the robot undergoing a turn maneuver. It can be seen that the wheels accelerate at a constant velocity, which makes sense considering the force applied is constant. Also, it can be seen that the pairs of wheels, one and three as well as two and four, are inverted. This makes sense since those pairs are opposites, as shown in Figure 3.3. The position of wheel two goes negative while wheel four goes positive. This means the velocity of two should be negative while four is positive.
Figure 4.1: The velocities vs positions using a constant force input
Figure 4.2 shows the front and rear path coordinates on straight-line trajectory. The track is created by using steering angles of zero and constant force applied to the drive wheels which means straight line motion should be produced. The coordinates almost overlap perfectly.
Figure 4.2: Front and Rear Coordinates on a straight line trajectory
The simulations from Figure 4.1 and Figure 4.2 both show model accuracy, albeit different facets of it. The accuracy displayed in Figure 4.1 is that of the wheels and how they
respond to force and torque inputs while the accuracy of Figure 4.2 is for the body to see if it consistently follows a straight line given inputs for that straight line.
There system is inherently unstable due to the nonlinear nature of the model. Another part of the instability is most likely due to the poor integration from the trapezoidal integrator used in the model. An accurate integrator, such as fourth order Runge-Kutta, as well as a controller would most likely clean up the response from Figure 4.1.
5. EXPERIMENTAL SIMULATION
While the experimental system was created, there was not time for experimental testing. All the sensors and actuators are operational, however, the overall programming architecture of the system was not completed. Further work needed to be done in several areas, including, increasing communication reliability between the PI and the ST microcontroller, creation of a P controller for the stepper motors, utilization of a more accurate sensor for the steering angle, data communication between the PI and a data logging laptop, communication between the PI and the GPS, and programming the SICK laser. However, much of the code is written as shown in, 0 and 0. Section 0 is the ‘h files’ for all the code used to program the robot while section 0 is the corresponding ‘c files’. The ‘main’ file can be found in, 0. This is the starting point for the code. The rest of the code branches out from that.
It was desired to simulate the system is a manner in which it was to be utilized, such as paths that may be common to mowing-lawn or moving snow. This means paths that are both grid like and with planned maneuvers throughout to represent obstacles from landscaping. Figure 5.1 shows a proposed trajectory. This model doesn’t have any obstacles built into it, but it does have turns that become tighter and tighter throughout the length of the path. This is meant to simulate the systems reaction to obstacles. The system may need to evade obstacles quickly which means a tight maneuver might need to be performed. The ability to track the path accurately in these circumstance is critical to mission performance.
Figure 5.1 could also be used for velocity testing. Once the speed limitations from the turning tests had been determined, tests that focus on the relationship between velocity and straight-line path tracking could be performed. The idea behind this is that the nonlinear model
should run at a certain ratio of the velocity. However, it may be that hardware limitations prevent that ratio from being reached. This may have an effect on path tracking.
Figure 5.1: A proposed trajectory to test the dynamic model of the 4WD4WS system. The
decreasing radius of the curves is meant to determine the path tracking accuracy under conditions of higher acceleration.
Thus, it should be determined what effect velocity has on path tracking performance and what amount of error is reasonable. It could be that a large velocity increase, resulting in a iteration to velocity ratio that is smaller than desirable, may result in an insignificant decrease in path tracking performance.
Tables could be made that show the error against velocity such that standards could be created for different tasks. Some tasks, like mowing of a professional baseball stadium, may require pinpoint accuracy, and a lower overall velocity, while other applications may be able to get by with less precision.
6. CONCLUDING REMARKS
This thesis has investigated the kinematic and dynamic modelling for a four-wheel independent-drive, four-wheel independent-steer robotic vehicle for the use as a consumer reconfigurable robotic system. The 4WD4WS structure was chosen to be a reconfigurable
system due to its ability to handle many unique environments, such as grass and snow, while also maximizing mission performance.
The developed dynamic model boasts very high fidelity when compared to current models. Kinematic relationships guarantee that the system will never drive into obstacles which further increases the models path-tracking potential. This model is designed for a robot of arbitrary width and length, to follow a path of arbitrary steering angles, in a vehicle with
arbitrary mass. The presents a system of incredible flexibility which means it can be incorporated into a wide variety of environments, including automotive, industrial, and consumer. This is in contrast to many models who limit their steering angles, use a vehicle of minimal mass, use a fixed frame size, or drive at small velocities in order to negate the effects of Newton’s second law. While these models work well for their limited application, they are not flexible.
It became quite clear during the derivation of the model that it was far more complex than initially thought. The original plan was to create the dynamic model and advanced controller in tandem with the experimental system. While the experimental system was successfully created, the dynamic model kept growing in complexity and scale. Thus, the new focus of the project became the completion of the system’s equations of motion. As discussed earlier, the dynamics are incredibly flexible. This is because we didn’t take shortcuts, as many do, in order to get a simpler system. As a result, the flexible system is like none before it. With these equations, a
follow up thesis would be in a very good spot to create a control algorithm to test with the completed experimental system.
6.1. Future Work
It is recommended that the current equations of motion be utilized to create a fully comprehensive control system that incorporates advanced control theory in conjunction with the advanced experimental vehicle. This would further prove the models effectiveness which may help the 4WD4WS structure be used in more mobile robots. It is also recommended that the control algorithm incorporate dynamic path planning. This would provide a system that would not only stay on the path but also avoid obstacles.
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APPENDIX
A.1. MATLAB Simulation Code